Central angle is the angle created by two arms through the facility of a circle together the vertex. The 2 arms type two radii that the one intersecting the arc that the circle at various points. Main angle helps to divide a circle into sectors. A slice that pizza is a great example of central angle. A pie graph is comprised of a variety of sectors and also helps to represent various quantities. A protractor is a straightforward example the a sector with a main angle of 180º. Central angle can additionally be identified as the angle created by an arc of the circle in ~ the center of the circle.

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1.Definition of main Angle
2. Central edge Theorem
3.How come Find central Angle?
4.Solved examples on central Angles
5.Practice inquiries on central Angles
6.FAQs on central Angles

Definition of central Angle

The angle subtended by one arc in ~ the center of a circle is the central angle. The radius vectors form the arms of the angle. In other words, an angle whose vertex is the facility of a circle with the 2 radii lines together its arms that crossing at two various points. When these 2 points space joined they type an arc. Main angle is the edge subtended through this arc at the center of the circle.



Here O is the facility of the circle, abdominal is the arc and, OA is a radius and also OB is one more radius of the circle.

Central Angle= (fracs imes 360^02 pi r)Here "s" is the size of the arc and also "r" is the radius of the circle. This is the formula for finding main angle in degrees. For finding the main angle in radians, we need to divide the arc length by the length of the radius that the circle.

Central angle Theorem

Theorem: The angle subtended by an arc in ~ the facility of the one is double the angle subtended by the at any other allude on the circumference of the circle.


The main angle theorem says that the central angle of a circle is dual the measure of the angle subtended by the arc in the various other segment the the circle.

∠AOB = 2 × ∠ACB 

Central edge = 2 × angle in various other segment

How come Find main Angle?

The central angle is the angle between any two radii of a circle. To discover the main angle we require to discover the arc size (which is the distance in between the 2 points that intersection that the the two radii) and the radius length. The actions given below shows how to calculate central angle in radians.

There are three simple steps to discover the main angle.

Identify the ends of the arc and the facility of the one (curve). AB is the arc of the circle and O is the center of the circle.


Join the end of the arc with the facility of the circle. Also, measure the length of the arc and the radius. Here abdominal is the size of the arc and also OA and OB space the radii of the circle.


Divide the size of the curve through the radius, to acquire the central angle. By using the formula displayed below, we will uncover the value of the central angle in radians. 

( extCentral Angle = dfrac extLength of the ArcRadius)

Important note

The main angle that a circle is measure in radian measure and sexagesimal measure.

The unit the radian measure is radians and also the unit of sexagesimal measure up is degrees.

Radian × (180/π) = Sexagesimal

Topics associated to central Angle

Check out these interesting posts to recognize about central angle and also its related topics.

Important topics

Solved Examples

Example 1: Sam measures the angle in a triangle through the aid of a protractor together 60º. Convert the angle into radian measure.Solution:

The offered angle of 60° is in sexagesimal measure.

Radian = π/180° × Sexagesimal Radian = π/180° × 60° 

Radian = π/3Therefore, the angle is π/3 radians.

Example 2: Larry drew a circle and cut it into 4 equal parts using 2 diameters. How deserve to you help Larry to measure up the central angle or enrolled angle of each component of the circle?Solution: 

Larry cuts the one into 4 equal parts.

Complete angle in a circle = 360°

Angle of each quadrant = 360°/4

= 90°Therefore, the main angle the a quadrant is 90°.

Example 3: Sally clues an arc of size 8 inches and also measures its central angle together 120 degrees. What is the radius that the arc?Solution: 

Radius that the arc = 8 inches

Central angle = 120°

Central edge = (length of arc × 360°)/(2 π × radius)

radius = (length that arc × 360°)/(2 π × central angle) 

radius =(8 × 360°) / (2 π × 120°)

radius = 12/π

Therefore, the radius is 12/π inches.

Example 4: Jim offers a compass to attract an arc of length 11 inches and also a radius of 7 inches. Without making use of a protractor, how deserve to Jim calculation the angle of this arc?Solution: 

Length the the arc = 11 inches

Radius of the arc = 7 inches

 Angle that the arc = (length of arc × 360°)/(2 π r)

Angle = (11 × 360°)/ (2 × 22/7 × 7)

Angle = 90°Therefore, the edge of the arc is 90°.

Example 5: George wants to develop a garden in the shape of a sector of radius 42 feet and having actually a central angle that 120 degrees. Calculate the area the the grass which is compelled to cover the garden. Solution: 

Given that the form of the garden is a sector.

Radius = 42 feet

Central angle = 120°

The area of the grass forced to covering the garden is the exact same as the area that the sector.

Area the the ar = θ/360° × π r2

Area =(120°/360°) × π × 422

Area = 1/3 × 22/7 × 42 × 42 

 Area = 22 × 2 × 42 

 Area = 1848 Therefore, the area of the sector is 1848 square feet.

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FAQs on main Angle

How perform You find the central Angle of an Arc?

To discover the main angle of one arc, connect the ends of the arc v the center of the circle utilizing the radius vectors. The angle between the two radii to represent the central angle that the arc.

What is the Difference between Reflex and also Convex Angles?

Both reflex angle and convex angle can be main angles of a circle. A reflex angle is higher than 180 degrees and also less 보다 360 degrees. A convex edge is less than 180 degrees.For a given arc that a circle, the sum of the convex angle and also reflex angle is equal to the complete angle. A complete angle is equal to 360 degrees. Convex angle + Reflex edge = complete angle

What Is central Angle definition in Geometry?

As per main angle meaning in geometry, the is the angle subtended through the arc of the circle in ~ the facility of the circle. The 2 radii make the arms of the angle.

How countless Degrees is the main Angle the a Circle?

The levels of a main angle is the angle made through the arc in ~ the facility of the circle.

What is The central Angle that a Curve?

The central angle the a curve is the angle subtended through it at the facility of the curve.

What is main Angle Theorem?

The central angle theorem claims that the angle subtended by an arc at the facility of the circle is dual the edge subtended at any point on the one of the circle.

How perform You Measure main Angle of a Circle?

The central angle the a one is measure in either levels or radians. That is measured through the aid of length of the arc and also length the the radius the the circle. The formula to measure central angle (in radians) = (Length of the arc)/(Length that the radius).

What is the main Angle made by a Semi-circle?

The central angle make by a semi-circle is 180°.

What is an enrolled Angle?

The edge subtended by one arc in ~ any point on the one is referred to as an enrolled angle.

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What is the difference Between main Angle and also Inscribed Angle?

Central edge is the angle subtended by one arc in ~ the center of a circle. Inscribed angle is an edge subtended by one arc in ~ any allude on the one of a circle.